William Jones and circles

William Jones worked on an important mathematical idea about circles. How has the artist cleverly shown you a circle in the portrait?

You will need some thread or string which you can cut easily then you can discover for yourself what interested William Jones about circles.

William Jones by William Hogarth, oil on canvas, 1740

Cut a piece of string that is exactly the same size as the diameter of this circle. Now find out how many times the diameter will fit around the circumference of the circle, by laying your string on the circumference. You will have a bit left over. Then fill in this sentence.

The diameter fits around the circumference of the circle

and a bit times.

Now find lots more circles and try them out. Fill in this sentence

The diameter always fits around the circumference of the circle

and a bit times.

Page 2

Extension

We can use Mr Jones's idea to help us find the area of a circle. Here is one way to find the area.

What is the area of this circle?

How did you find it?

What did you do about all the bits that were not whole squares?

Is this a very accurate way to measure the area of a circle?

Page 3

To do the next stage you will need a circle of sticky paper and scissors.

Fold your sticky paper circle so you have 8 pieces this shape.

Cut them out neatly.

Now stick them together like this, using all your pieces.

The shape you have now made should look roughly like a rectangle. Does it?

How do you find the area of a rectangle?

Now use a ruler to measure your rectangle and then work out its area (remember, your answer will be in square centimetres)

Do you think this is a more accurate way to find the area of a circle than the counting squares way? Why is it still not totally accurate?

Page 4

Further Extension

Let's now look at your rectangle and see what happened to the parts of the circle when we cut it up.

this piece is called the circumference

this piece is half the diameter -

we call it the radius

So if we multiply half the circumference by the radius (that's half the diameter) we will get the area. Let's now think again about what we can call the circumference. It is two lots of the radius 3 and a bit times over.

this will go ground 3 and a bit times

To get the area, we need to multiply half the circumference by the radius - so we have

half the circumference x radius = radius x 3 and a bit x radius

It's easier to work it out as: radius x radius x 3 and a bit

Page 5

To do the maths you will now need a calculator. The "bit" is about 0.14 so 3 and a "bit" is 3.14

Now to find the area of any circle, all you need to do is to measure the radius, multiply it by itself and then by 3.14.

This method is a more accurate way to find the area of a circle than either the counting squares or cutting sticky paper way. Are there bits of the circle we haven't taken into account?

Remember we wouldn't be able to work out the area of a circle like this without knowing that the diameter of a circle fits round the circumference 3.14 times. Because this is quite long number to keep writing down, William Jones gave it a sign - the Greek letter which is said "pye".

Some calculators have the sign on one button - if you press it you will get 3.14